# Interpolated FIR filter (from Oppenheim and Schafer's Discrete-Time Signal Processing, 3rd ed)

[from Discrete-time Signal Processing by Oppenheim and Schafer, 3rd ed., p.196]

Two questions:

In this context, the filter with system function represented by Eq. (103) is called an interpolated FIR filter. This is because the corresponding impulse response can be seen to be the convolution of $$h_1[n]$$ with the second impulse response expanded by $$M_1$$.

1. This explanation as to why Eq. (103) is called an interpolated FIR filter is still not clear for me.
2. Why is the impulse response given by Eq (104) FIR?

1. The reason why "interpolated FIR filter" is an appropriate name is because the filter $$h_2[n]$$ has $$M_1-1$$ zeros between its (non-zero) filter taps, and the convolution with the filter $$h_1[n]$$ interpolates those $$M_1-1$$ values.
2. Eq. $$(104)$$ generally only represents an FIR filter if both $$h_1[n]$$ and $$h_2[n]$$ are FIR filters.